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January, 1992 Fusions of a Probability Distribution
J. Elton, T. P. Hill
Ann. Probab. 20(1): 421-454 (January, 1992). DOI: 10.1214/aop/1176989936


Starting with a Borel probability measure $P$ on $X$ (where $X$ is a separable Banach space or a compact metrizable convex subset of a locally convex topological vector space), the class $\mathscr{F}(P)$, called the fusions of $P$, consists of all Borel probability measures on $X$ which can be obtained from $P$ by fusing parts of the mass of $P$, that is, by collapsing parts of the mass of $P$ to their respective barycenters. The class $\mathscr{F}(P)$ is shown to be convex, and the ordering induced on the space of all Borel probability measures by $Q \prec P$ if and only if $Q \in \mathscr{F}(P)$ is shown to be transitive and to imply the convex domination ordering. If $P$ has a finite mean, then $\mathscr{F}(P)$ is uniformly integrable and $Q \prec P$ is equivalent to $Q$ convexly dominated by $P$ and hence equivalent to the pair $(Q, P)$ being martingalizable. These ideas are applied to obtain new martingale inequalities and a solution to a cost-reward problem concerning optimal fusions of a finite-dimensional distribution.


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J. Elton. T. P. Hill. "Fusions of a Probability Distribution." Ann. Probab. 20 (1) 421 - 454, January, 1992.


Published: January, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0747.60009
MathSciNet: MR1143430
Digital Object Identifier: 10.1214/aop/1176989936

Primary: 60B05
Secondary: 60G42

Keywords: balayage , convex domination , dilation , Fusion of a probability , Hardy-Littlewood maximal function , majorization , martingalizable

Rights: Copyright © 1992 Institute of Mathematical Statistics


Vol.20 • No. 1 • January, 1992
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